Incidentally, getting back to the original question: Back when case meant something in English and we actually marked it in nouns different prepositions took different cases in their noun objects. We have dumped all case markings in nouns except the genitive. So, the one case where it is shown we think is abnormal. Really, it's all the instances where it's not shown that are the abnormal ones.

I have been looking back over this discussion, and there is one possible explanation that no-one has mentioned.

"A friend of my sister's" implies that my sister has two or more friends and we are referring to one of them. So we mean "a friend of [= from among] my sister's friends".

"Of" here means not "belonging to my sister's friends" but "part of the group of her friends". This is why we cannot say "the friend of my sister's" – there cannot be a group if she has only one friend.

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